Monad transformer = thing that can transform a monad into another one with additional primitive operations. Monads are more or less the Haskell way of achieving something akin to overloading the semicolon in imperative languages. In imperative languages the meaning of the semicolon is baked into the language. It seems natural and stupid to think about what it means, normally it's just "do this, changing the program state by assigning some values to some variables, then do this with this new state" but if you think harder, it can also mean "do this and skip the rest of the loop" (if the first statement is a break) or "do this and then skip a bunch of levels up the call stack until you find an appropriate handler, unwinding the stack in the process" (if it's a throw). Haskell doesn't have any of this baked in, but in a way allows you to define the semantics of the semicolon yourself.

So monad transformers then allow you to build up more complicated semicolon semantics by combining simpler ones (and getting the full set of their primitive operations, such as assignment, break or throw statements in the previous examples).

Logic programming, in its simplest form, is concerned with deducing "goal" facts of the form C(X), from a bunch of rules of the form "A(X) and B(X) and ... imply C(X)" and other facts of the form "A(X)". One way you can do this is look for all the rules with your goal fact as their conclusion, then look how you can derive their premises, and so on. Which essentially boils down to a backtracking search.

So what Kiselyov et al did was to implement some primitive operations and overload the semicolon in a way which makes it easy to perform a backtracking search. Or more precisely, since it's a monad transformer, they figured out a way to add these primitives to any set of existing ones (such as assignment primitives for instance). Their implementation also provides some interesting backtracking operations which can be tricky to implement (the aforementioned fair disjunctions, negation as failure and pruning). And it is efficient since it's based on continuations (which are just functions of a certain format), as compared to other approaches which first have to represent the target program as data ("reify" it), then define an interpreter for that data and finally run the interpreter.

Better?